MathDB

2008 JBMO Shortlist

Part of JBMO ShortLists

Subcontests

(11)
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p(x^2 - y^2) = (p^2- 1)xy , |x - 1|+ |y| = 1

Let the real parameter pp be such that the system {p(x2−y2)=(p2−1)xy∣x−1∣+∣y∣=1\begin{cases} p(x^2 - y^2) = (p^2- 1)xy \\ |x - 1|+ |y| = 1 \end{cases} has at least three different real solutions. Find pp and solve the system for that pp.

2n integers arbitrarily assigned to 2n numbered boxes

Integers 1,2,...,2n1,2, ...,2n are arbitrarily assigned to boxes labeled with numbers 1,2,...,2n1, 2,..., 2n. Now, we add the number assigned to the box to the number on the box label. Show that two such sums give the same remainder modulo 2n2n.

a_{n+1} = a_n+s(a_n), whereas s(a)=sum of digits of a

Let s(a)s(a) denote the sum of digits of a given positive integer a. The sequence a1,a2,...,an,...a_1, a_2,..., a_n, ... of positive integers is such that an+1=an+s(an)a_{n+1} = a_n+s(a_n) for each positive integer nn. Find the greatest possible n for which it is possible to have an=2008a_n = 2008.
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3 positive reals w product 1,sum their pairwise product sums

Kostas and Helene have the following dialogue: Kostas: I have in my mind three positive real numbers with product 11 and sum equal to the sum of all their pairwise products. Helene: I think that I know the numbers you have in mind. They are all equal to 11. Kostas: In fact, the numbers you mentioned satisfy my conditions, but I did not think of these numbers. The numbers you mentioned have the minimal sum between all possible solutions of the problem. Can you decide if Kostas is right? (Explain your answer).

2008 JBMO Shortlist G2

For a fixed triangle ABCABC we choose a point MM on the ray CACA (after AA), a point NN on the ray ABAB (after BB) and a point PP on the ray BCBC (after CC) in a way such that AM−BC=BN−AC=CP–ABAM -BC = BN- AC = CP – AB. Prove that the angles of triangle MNPMNP do not depend on the choice of M,N,PM, N, P .

set of rationals not multiple of an $n$-th power of a prime

Let n≥2n \ge 2 be a fixed positive integer. An integer will be called "nn-free" if it is not a multiple of an nn-th power of a prime. Let MM be an infi nite set of rational numbers, such that the product of every nn elements of MM is an nn-free integer. Prove that MM contains only integers.
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0<a,b,c,d<1 => 1 + ab + bc + cd + da + ac + bd > a+b+c+d

If the real numbers a,b,c,da, b, c, d are such that 0<a,b,c,d<10 < a,b,c,d < 1, show that 1+ab+bc+cd+da+ac+bd>a+b+c+d1 + ab + bc + cd + da + ac + bd > a + b + c + d.

2008 JBMO Shortlist G6

Let ABCABC be a triangle with ∠A<90o\angle A<{{90}^{o}} . Outside of a triangle we consider isosceles triangles ABEABE and ACZACZ with bases ABAB and ACAC, respectively. If the midpoint DD of the side BCBC is such that DE⊥DZDE \perp DZ and EZ=2⋅EDEZ = 2 \cdot ED, prove that ∠AEB=2⋅∠AZC\angle AEB = 2 \cdot \angle AZC .

for n, exists p so that p|n and f(n) = f(n/p)-f(p)

Let f:N→Rf : N \to R be a function, satisfying the following condition: for every integer n>1n > 1, there exists a prime divisor pp of nn such that f(n)=f(np)−f(p)f(n) = f \big(\frac{n}{p}\big)-f(p). If f(22007)+f(32008)+f(52009)=2006f(2^{2007}) + f(3^{2008}) + f(5^{2009}) = 2006, determine the value of f(20072)+f(20083)+f(20095)f(2007^2) + f(2008^3) + f(2009^5)
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&Sigma;x_i \cdot &Sigma;1/x_i &gt;= 4 (&Sigma;&chi;_i/(x_ix_j+1))^3 , i=1,2,3

Show that (x+y+z)(1x+1y+1z)≥4(xxy+1+yyz+1+zzx+1)2(x + y + z) \big(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\big) \ge 4 \big(\frac{x}{xy+1}+\frac{y}{yz+1}+\frac{z}{zx+1}\big)^2 , for all real positive numbers x,yx, y and zz.

2008 JBMO Shortlist G8

The side lengths of a parallelogram are a,ba, b and diagonals have lengths xx and yy. Knowing that ab=xy2ab = \frac{xy}{2}, show that (a,b)=(x2,y2)\left( a,b \right)=\left( \frac{x}{\sqrt{2}},\frac{y}{\sqrt{2}} \right) or (a,b)=(y2,x2)\left( a,b \right)=\left( \frac{y}{\sqrt{2}},\frac{x}{\sqrt{2}} \right).

1 square is sum of 3 squares in 2 different ways

Let a,b,c,d,e,fa, b, c, d, e, f are nonzero digits such that the natural numbers abc‾,def‾\overline{abc}, \overline{def} and abcdef‾\overline{abcdef } are squares. a) Prove that abcdef‾\overline{abcdef} can be represented in two different ways as a sum of three squares of natural numbers. b) Give an example of such a number.
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1-1024, deleting 4k+3 numbers, successively for 5 times

Consider an integer n≥4n \ge 4 and a sequence of real numbers x1,x2,x3,...,xnx_1, x_2, x_3,..., x_n. An operation consists in eliminating all numbers not having the rank of the form 4k+34k + 3, thus leaving only the numbers x3.x7.x11,...x_3. x_7. x_{11}, ...(for example, the sequence 4,5,9,3,6,6,1,84,5,9,3,6, 6,1, 8 produces the sequence 9,19,1). Upon the sequence 1,2,3,...,10241, 2, 3, ..., 1024 the operation is performed successively for 55 times. Show that at the end only one number remains and fi nd this number.

2008 JBMO Shortlist G9

Let OO be a point inside the parallelogram ABCDABCD such that ∠AOB+∠COD=∠BOC+∠AOD\angle AOB + \angle COD = \angle BOC + \angle AOD. Prove that there exists a circle kk tangent to the circumscribed circles of the triangles △AOB,△BOC,△COD\vartriangle AOB, \vartriangle BOC, \vartriangle COD and △DOA\vartriangle DOA.

(4a + p)/b+\(4b + p)/a , a^2/b+b^2/a \in Z

Let pp be a prime number. Find all positive integers aa and bb such that: 4a+pb+4b+pa\frac{4a + p}{b}+\frac{4b + p}{a} and a2b+b2a \frac{a^2}{b}+\frac{b^2}{a} are integers.
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a,b,c&gt;0 , abc=1 =&gt; &Pi; (ab + bc +1/ca)&gt;=&Pi;(1+2a)

Let a,ba, b and cc be positive real numbers such that abc=1abc = 1. Prove the inequality (ab+bc+1ca)(bc+ca+1ab)(ca+ab+1bc)≥(1+2a)(1+2b)(1+2c)\Big(ab + bc +\frac{1}{ca}\Big)\Big(bc + ca +\frac{1}{ab}\Big)\Big(ca + ab +\frac{1}{bc}\Big)\ge (1 + 2a)(1 + 2b)(1 + 2c).

2008 JBMO Shortlist G7

Let ABCABC be an isosceles triangle with AC=BCAC = BC. The point DD lies on the side ABAB such that the semicircle with diameter BDBD and center OO is tangent to the side ACAC in the point PP and intersects the side BCBC at the point QQ. The radius OPOP intersects the chord DQDQ at the point EE such that 5⋅PE=3⋅DE5 \cdot PE = 3 \cdot DE. Find the ratio ABBC\frac{AB}{BC} .

Minimum value

Determine the minimum value of prime p>3p> 3 for which there is no natural number n>0n> 0 such that 2n+3n≡0(modp)2^n+3^n\equiv 0\pmod{p} .